L(s) = 1 | + i·3-s − 2.47i·5-s + 2.55·7-s − 9-s + 0.669i·11-s + 4.08i·13-s + 2.47·15-s + 6.44·17-s − 6.44i·19-s + 2.55i·21-s − 2.82·23-s − 1.11·25-s − i·27-s + 4.35i·29-s + 6.55·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.10i·5-s + 0.966·7-s − 0.333·9-s + 0.201i·11-s + 1.13i·13-s + 0.638·15-s + 1.56·17-s − 1.47i·19-s + 0.558i·21-s − 0.589·23-s − 0.223·25-s − 0.192i·27-s + 0.808i·29-s + 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.192001490\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.192001490\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 + 2.47iT - 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 - 0.669iT - 11T^{2} \) |
| 13 | \( 1 - 4.08iT - 13T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 + 6.44iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 4.35iT - 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 - 3.85iT - 37T^{2} \) |
| 41 | \( 1 + 0.788T + 41T^{2} \) |
| 43 | \( 1 + 0.550iT - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 3.64iT - 53T^{2} \) |
| 59 | \( 1 + 5.65iT - 59T^{2} \) |
| 61 | \( 1 - 6.20iT - 61T^{2} \) |
| 67 | \( 1 - 2.99iT - 67T^{2} \) |
| 71 | \( 1 + 5.11T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 - 6.31T + 79T^{2} \) |
| 83 | \( 1 - 0.907iT - 83T^{2} \) |
| 89 | \( 1 - 6.31T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.633046789769704549102848640651, −8.233041575434280971174682693243, −7.31505666790400971083933452706, −6.39221018254865142819835083978, −5.28333566830976844581456278970, −4.84581917771650511012431022684, −4.29178600184128963978245123993, −3.17738699123492055549168626944, −1.90628752906866056635895765672, −0.928196709972992041793906517175,
0.967129478159077582811312520416, 2.06946346147649236706178109143, 3.05236433251727641175517067444, 3.74049788703087802830839835185, 5.02386346447950319029565010852, 5.84298727536451012185844115050, 6.33307368301253062200441842189, 7.44683267047378615867486286055, 7.967673174072044158143295842785, 8.212233894617340227905806323890